Advertisement

Numerische Mathematik

, Volume 100, Issue 1, pp 71–89 | Cite as

On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations

  • Volker GrimmEmail author
Article

Summary.

This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions.

Keywords

Differential Equation Mathematical Method General Equation Linear Time Linear Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ascher, U.M., Reich, S.: On some difficulties in integrating highly oscillatory Hamiltonian systems. In: Proc. Computational Molecular Dynamics, Springer Lecture Notes, 1999, pp. 281–296Google Scholar
  2. 2.
    Cohen, D., Hairer, E., Lubich, Ch.: Modulated fourier expansions of highly oscillatory differential equations. Foundations of Comput. Maths. 3, 327–450 (2003)CrossRefGoogle Scholar
  3. 3.
    García-Archilla, B., Sanz-Serna, J., Skeel, R.: Long-time-step methods for oscillatory differential equations. SIAM J. Sci. Comput. 30(3), 930–963 (1998)Google Scholar
  4. 4.
    Grimm, V.: Exponentielle Integratoren als Lange-Zeitschritt-Verfahren für oszillatorische Differentialgleichungen zweiter Ordnung, PhD thesis, Mathematisches Institut, Universität Düsseldorf, Germany, 2002Google Scholar
  5. 5.
    Grubmüller, H.: Dynamiksimulation sehr großer Makromoleküle auf einem Parallelrechner, PhD thesis, Physik-Dept. der Tech. Univ. München, 1994Google Scholar
  6. 6.
    Hairer, E., Lubich, C.: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38, 414–441 (2000)CrossRefGoogle Scholar
  7. 7.
    Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration. Springer-Verlag, 2002Google Scholar
  8. 8.
    Hochbruck, M., Lubich, Ch.: A Bunch of Time Integrators for Quantum/Classical Molecular Dynamics. In: P. Deuflhard, J. Hermans, B. Leimkuhler, A.E. Mark, S. Reich, R.D. Skeel (eds.), Computational Molecular Dynamics: Challenges, Methods, Ideas, volume 4 of Lecture Notes in Computational Science and Engineering, Springer-Verlag, Berlin, 1997, pp. 421–432Google Scholar
  9. 9.
    Hochbruck, M., Lubich, Ch.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)CrossRefGoogle Scholar
  10. 10.
    Hochbruck, M., Lubich, Ch.: A Gautschi-type method for oscillatory second-order differential equations. Numer. Math. 83, 403–426 (1999)CrossRefGoogle Scholar
  11. 11.
    Hochbruck, M., Lubich, Ch., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. 19, 1552–1574 (1998)CrossRefGoogle Scholar
  12. 12.
    Iserles, A.: Think globally, act locally: solving highly-oscillatory ordinary differential equations. Appld. Num. Anal. 43, 145–160 (2002)Google Scholar
  13. 13.
    Izaguirre, J.A., Reich, S., Skeel, R.D.: Longer time steps for molecular dynamics. J. Chemical Phys. 110(20), 9853–9864 (1999)CrossRefGoogle Scholar
  14. 14.
    Petzold, L., Jay, L., Yen, J.: Numerical Solution of Highly Oscillatory Ordinary Differential Equations. Acta Numerica 6, 437–484 (1997)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Mathematics DepartmentLa Trobe UniversityMelbourneAustralia

Personalised recommendations